Imaging techniques typically comprise detecting a signal from an object and constructing an image based on the detected signal. The detected signal may include any detectable datum from the sample, such as an electromagnetic signal from any frequency range, a magnetic signal, an ionization signal, heat, particles (electron, proton, neutron, etc.), or the like.
The imaged object may comprise any portion of a living organism (e.g., human or animal) or a nonliving object. For example, the portion may comprise an internal or an external portion, or may comprise the entire internal or external portion of the object. There are a wide variety of techniques for imaging of the object. Examples of imaging techniques include, but are not limited to: computed tomography (CT), positron emission tomography (PET), single-photon emission computed tomography (SPECT), magnetic resonance imaging (MRI), electron paramagnetic resonance imaging (EPRI), wave imaging (such as phase contrast imaging, thermacoustic imaging, and thermooptical imaging), and particle imaging. Further, various imaging techniques may be combined. For example, CT imaging and PET imaging may be combined to generate an image.
CT is an X-ray procedure in which the X-ray beam may move around the object, taking pictures from different angles. These images may be combined by a computer to produce a cross-sectional picture of the inside of the object. PET is a diagnostic imaging procedure that may assess the level of metabolic activity and perfusion in various organ systems of an object, such as a human body. A positron camera (tomograph) may be used to produce cross-sectional tomographic images, which may be obtained from positron emitting radioactive tracer substances (radiopharmaceuticals), such as 2-[F-18] Fluoro-D-Glucose (FDG), that may be administered intravenously to the object. SPECT scans and PET scans are part of the nuclear imaging family. The SPECT scan is capable of revealing information about the object, such as blood flow to tissue. For example, radionuclide may be given intravenously, with the tissues absorbing the radionuclides (diseased tissue absorbs at a different rate), and the rotating camera picking up images of these particles, which may then be transferred to a computer. The images may be translated onto film as cross sections and can be viewed in a 3-D format. Moreover, MRI and EPRI are imaging techniques that use a magnetic field and radiofrequency radiation to generate information, such as anatomical information.
In certain instances, the images may be generated using the exemplary imaging techniques discussed above from full knowledge of their linear transforms. However, in many practical situations, one may have access only to fractions of such measurements and thus have limited (instead of full) knowledge of the linear transforms. Thus, in various forms of imaging, including tomography, one of the main issues for image reconstruction centers on data sufficiency and on how to estimate an image (such as a tomographic image) when the projection data are not theoretically sufficient for exact image reconstruction. Insufficient data problems occur quite frequently because of practical constraints due to the imaging hardware, scanning geometry, or ionizing radiation exposure. The insufficient data problem may take many forms. For example, one type of the insufficient data problem derives from sparse samples, such as attempting to reconstruct an image from projection data at few views. Another example of an imperfect scanning data situation comprises limited angular range of the object to be imaged. Still another example comprises gaps in the projection data caused by bad detector bins, metal within the object, etc. In each of these three examples, the projection data are not sufficient for exact reconstruction of tomographic images and application of standard analytic algorithms, such as filtered back-projection (FBP), may lead to conspicuous artifacts in reconstructed images.
Methodologies have been proposed attempting to overcome data insufficiency in tomographic imaging. The methodologies follow one of two approaches. The first approach includes interpolating or extrapolating the missing data regions from the measured data set, followed by analytic reconstruction. Such an approach may be useful for a specific scanning configuration, imaging a particular object. However, this approach is, very limited, and is not applicable generally to the data insufficiency problem. The second approach employs an iterative methodology to solve the data model for images from the available measurements. Iterative methodologies have been used for tomographic image reconstruction. These methodologies differ in the constraints that they impose on the image function, the cost function that they seek to minimize, and the actual implementation of the iterative scheme.
Two iterative methodologies used for tomographic imaging include: (1) the algebraic reconstruction technique (ART); and (2) the expectation-maximization (EM) methodology. For the case where the data are consistent yet are not sufficient to determine a unique solution to the imaging model, the ART methodology finds the image that is consistent with the data and minimizes the sum-of-squares of the image pixel values. The EM methodology applies to positive integral equations, which is appropriate for the CT-imaging model, and seeks to minimize the Kullback-Liebler distance between the measured data and the projection of the estimated image. The EM methodology has the positivity constraint built into the algorithm, so that it is relatively unaffected by data inconsistencies introduced by signal noise. However, the EM methodology is limited in its ability to solve the data insufficiency problem.
For specific imaging problems, an accurate iterative scheme may be derived for the imperfect sampling problem by making a strong assumption on the image function. For example, in the specific example of reconstruction of blood vessels from few-view projections, one can assume that the 3D blood-vessel structure is sparse. It is possible to design an effective iterative algorithm that seeks a solution from sparse projection data. This can be accomplished by minimizing the l1-norm of the image constrained by the fact that the image yields the measured projection data. The l1-norm of the image is simply the sum of the absolute values of the image pixel values, and its minimization subject to linear constraints leads to sparse solutions. Again, this solution to the sparse data problem only addresses a very specific type of imaging.
Still another methodology uses total variation (TV) for recovering an image from sparse samples of its Fourier transform (FT). TV has been utilized in image processing for denoising of images while preserving edges. In this methodology, the optimization program of minimizing the image TV was investigated under the constraint that the FT of the image matches the known FT samples. This optimization program may satisfy an “exact reconstruction principle” (ERP) for sparse data. Specifically, if the number of FT samples is twice the number of non-zero pixels in the gradient image, then this optimization program can yield a unique solution, which is in fact the true image for almost every image function. The algorithm for FT inversion from sparse samples was applied to image reconstruction from 2D parallel-beam data at few-views. The use of the FT-domain TV algorithm (FT-TV) to address the 2D parallel-beam problem is only possible because of the central slice theorem, which links the problem to FT inversion. However, the FT-TV methodology is limited to imaging using a parallel-beam and cannot be applied to image reconstruction for divergent-beams, such as fan-beam and cone-beam CT. This is because the FT-TV relies on the central slice theorem to bring the projection data into the image's Fourier space. Therefore, there is a need to reconstruct images from few view or limited angle data generated from divergent beams.